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The order boundedness of band preserving operators on uniformly complete vector lattices

Published online by Cambridge University Press:  24 October 2008

P. T. N. McPolin  and
A. W. Wickstead
P. T. N. McPolin
Affiliation:
The Queen's University of Belfast, BT7 1NN
A. W. Wickstead
Affiliation:
The Queen's University of Belfast, BT7 1NN
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Extract

1. Introduction. A linear operator T on a vector lattice is band preserving if x⊥ y implies Tx ⊥ y. Much is known about the order bounded band preserving operators on an Archimedean vector lattice. The collection of all of these forms an Abelian algebra under composition and a vector lattice for the operator order (see [7], [8] and [13] amongst others). Very little appears to be known about band preserving operators which are not order bounded apart from some isolated examples ([11], [13], [1] and [17]) and some non-existence results ([11] and [1]).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 3 , May 1985 , pp. 481 - 487
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

[1]Abramovic, J. A., Veksler, A. I. and Koldunov, A. V.. On operators preserving disjointness. Soviet Math. Dokl. 20 (1979), 1089–1093.Google Scholar
[2]Abramovic, J. A.. Multiplicative representation of the operators preserving disjointness. Indag. Math. 45 (1983).Google Scholar
[3]Aliprantis, C. D. and Burkinshaw, O.. Locally solid Riesz spaces (Academic Press, 1978).Google Scholar
[4]Arendt, W.. Spectral properties of Lamperti operators. Indiana Univ. Math. J. 32 (1983), 199–215.CrossRefGoogle Scholar
[5]Bernau, S. J.. Lateral and Dedekind completion of Archimedean lattice groups. J. London Math. Soc. 12 (1976), 320–322.CrossRefGoogle Scholar
[6]Bernau, S. J.. Orthomorphisms of archimedean vector lattices. Math. Proc. Cambridge Philos. Soc. 89 (1981), 119–128.CrossRefGoogle Scholar
[7]Bigard, A. and Keimel, K.. Sur les endomorphismes conservant les polaires d'un groupe réticulé Archimédien. Bull. Soc. Math. France 97 (1969), 381–398.CrossRefGoogle Scholar
[8]Conrad, P. F. and Diem, J. E.. The ring of polar preserving endomorphisms of an abelian lattice ordered group. Illinois J. Math. 15 (1971), 224–240.CrossRefGoogle Scholar
[9]Dixmier, J.. Sur certains espaces considérés par M. H. Stone. Summa Bras. Math. 2 (1951), 151–182.Google Scholar
[10]Fremlin, D. G.. Inextensible Riesz spaces. Math. Proc. Cambridge Philos. Soc. 77 (1975), 71–90.CrossRefGoogle Scholar
[11]Luxemburg, W. A. J.. Some Aspects of the Theory of Riesz Spaces (Univ. Arkansas Lecture Notes in Maths., 4, 1979).Google Scholar
[12]Luxemburg, W. A. J. and Zaanen, A. C.. Riesz Spaces I (North-Holland 1971).Google Scholar
[13]Meyer, M.. Quelques propriétés des homomorphismes d'espaces vectoriels réticulés. Equipe d'Analyse, E.R.A. 294 (Université de Paris VI, Paris, 1979).Google Scholar
[14]Pagter, B. De. A note on disjointness preserving operators. Proc. Amer. Math. Soc. 90 (1984), 543–549.CrossRefGoogle Scholar
[15]Semadeni, Z.. Banach Spaces of Continuous Functions (Polish Scientific Publishers, 1971).Google Scholar
[16]Veksler, A. I. and Geiler, V. A.. Order and disjoint completeness of linear partially ordered spaces. Siberian Math. J. 13 (1972), 30–35.CrossRefGoogle Scholar
[17]Wickstead, A. W.. Extensions of orthomorphisms. J. Austral. Math. Soc. 29 (1980), 87–98.CrossRefGoogle Scholar